Question

using matlab help to answer #2 please show steps in creating code

2. The energy of the mass-spring system is given by the sum of the kinetic energy and the potential energy. In the absence of damping, the energy is conserved (a) Add commands to LAB05ex1 to compute and plot the quantity E-m k2 as a function of time. What do you observe? (pay close attention to the y-axis scale and, if necessary, use ylim to get a better graph). Ilude at least one plot. Does the graph confirm the fact that the energy is conserved? (b) Show analytically that = 0. (Note that this proves that the energy is constant) (c) Add commands to LABO5ex1 to plot v vs y (phase plot). Include the plot. Does the curve ever get close to the origin? Why or why not? What does that mean for the mass-spring system? Mass-Spring System with Damping When the movement of the mass is damped due to viscous effects (e.g., the mass moves in a cylinder containing oil, situation (d) an additional term proportional to the velocity must be THIS CONTENT IS PROTECTED AND MAY NOT BE SHARED, UPLOADED, SOLD OR DISTRIBUTED 2019 vS Copyrighto School of Mathematical and Statist cal Sciences, Ariaona State University added. The resulting equation becomes by setting pThe program LABO5ex1 is updated by modifying the function f. Here we set c= 1 . LAB05exla.m clear all; m-1; % mass [kg] k=4; % spring constant [N/m] c-1; % friction coefficient [Ns/m] omega0sqrt (k/m) pc/(2*m); у0--0 . 8 ; v0--0.3; % initial conditions [t,Y] = ode45 (0f, [0,10], [yo,vo],[], omegao, p); % solve for 0

Answer 1

Please find attached the copy of the matlab script LAB05ex1a.m below with parts (a) and (c) completed in the script along with the plots.

(a) On line 9 added Command 2 On line 1o appended the plot comno with theollbuing parameters t, E, The observaticn is that the energy cure is a hori 2ontal line l.e. energy (ie. total- . energy) is constant w.rt time and do not change with time. Hence the graph conçirms the pact that energy is conserved. ー (b) Since, E - mv2+ 1 ky2 2 Dipfeventiate Ewgive s dt 2 From equation of motion

dt dt dt Hence proved, (c) On line 15 added the command to plob Figure (2) Plot (y v) the curve hever gets cloce to the origin ecause encrgy is conserved so the motion never dampens le. ampitude op the motion does not decreases with time Hence this Spring-mass system has its energ u Conserved.

Name 1-clear; close all; clc; 2-m=1; 109xl double omegao 0.5000 109xl double 109x1 double -0.300O 109xl double 109x2 double 0.8000 [t, Y] ode45(@f, [0, 10], [yo, v0], [], p); 7_ omegad, VO yo 10-figure (1) plot( t, y, 'bo-', t. v, 'r+-, t, E, 'g) 11-xlabel ('t') 12 egend ('y'. 'V', 'E 13- grid on; axis tight 14 15-figure (2) plot( y, v ) 16- xlabel ( ): ylabel ( V' ) ; 17-grid on; axis square; 18 19 function dYdt =f(t, Y, omegao, p) 22 -end Command Window New to MATLAB? See resources for Getting Started. Ln6 Col 22

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